home/regression-approach.md

 ---
-title: regression approach - powerlaw
+title: regression approach and tolerance bands - power law
 ---
 
+
 ## 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).
@@ -75,13 +76,13 @@ 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, it also aids in faster convergence and accuracy.
 
 ```math
-X_{d,s} = \frac{X_d - X_{\textrm{d,min}}}{X_{\textrm{d,max}} - X_{\textrm{d,min}}}, \quad
-Y_{d,s} = \frac{Y_d - Y_{\textrm{d,min}}}{Y_{\textrm{d,max}} - Y_{\textrm{d,min}}}
+X_{\textrm{d,s}} = \frac{X_\textrm{d} - X_{\textrm{d,min}}}{X_{\textrm{d,max}} - X_{\textrm{d,min}}}, \quad
+Y_{\textrm{d,s}} = \frac{Y_\textrm{d} - Y_{\textrm{d,min}}}{Y_{\textrm{d,max}} - Y_{\textrm{d,min}}}
 ```
 
 ```math
-X_{p,s} = \frac{X_p - X_{\textrm{d,min}}}{X_{\textrm{d,max}} - X_{\textrm{d,min}}}, \quad
-Y_{p,s} = \frac{Y_p - Y_{\textrm{d,min}}}{Y_{\textrm{d,max}} - Y_{\textrm{d,min}}}
+X_{\textrm{p,s}} = \frac{X_\textrm{p} - X_{\textrm{d,min}}}{X_{\textrm{d,max}} - X_{\textrm{d,min}}}, \quad
+Y_{\textrm{p,s}} = \frac{Y_\textrm{p} - Y_{\textrm{d,min}}}{Y_{\textrm{d,max}} - Y_{\textrm{d,min}}}
 ```
 
 where the lower index s is an abbreviation for scaled.
@@ -94,37 +95,37 @@ The purpose is to sort the difference between the prediction and the data and de
 ![quantile_exmaple_error](uploads/f6950315da8dfc9c3d3054c0dfd95cb5/quantile_exmaple_error.png){width=35%}
 </div>
 
-Compute the error/difference between the model and the data 
+**a)** Compute the error/difference between the model and the data 
 
 ```math
 E = Y_{\textrm{p,s}} - Y_{\textrm{d,s}}
 ```
 
-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.
+**b)** 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
-E = E\left[E > 0\right], \quad X_{d,s} = X_{d,s}[E>0], \quad Y_{d,s} = Y_{d,s}[E>0]
+E = E\left[E > 0\right], \quad X_{\textrm{d,s}} = X_{\textrm{d,s}}[E>0], \quad Y_{\textrm{d,s}} = Y_{\textrm{d,s}}[E>0]
 ```
 
-Sort the data based on the error
+**c)** Sort the data based on the error
  
 ```math
 E = E\left[\textrm{argsort}\left(E\right)\right],
-\quad X_{d,s} = X_{d,s}\left[\textrm{argsort}\left(E\right)\right],
-\quad Y_{d,s} = Y_{d,s}\left[\textrm{argsort}\left(E\right)\right]
+\quad X_{\textrm{d,s}} = X_{\textrm{d,s}}\left[\textrm{argsort}\left(E\right)\right],
+\quad Y_{\textrm{d,s}} = Y_{\textrm{d,s}}\left[\textrm{argsort}\left(E\right)\right]
 ```
-Extract the data within the desired quantile range
+**d)** Extract the data within the desired quantile range
 ```math
 E = E\left[:-I_{\alpha}\right],
-\quad X_{d,s} = X_{d,s}\left[:-I_{\alpha}\right],
-\quad Y_{d,s} = Y_{d,s}\left[:-I_{\alpha}\right]
+\quad X_{\textrm{d,s}} = X_{\textrm{d,s}}\left[:-I_{\alpha}\right],
+\quad Y_{\textrm{d,s}} = Y_{\textrm{d,s}}\left[:-I_{\alpha}\right]
 ```
 **Step 6** <br>
 Renormalize the data.
 
 ```math
-X_{\Gamma} = X_{d,s} \left(X_{\textrm{d,max}} - X_{\textrm{d,min}}\right) + X_{\textrm{d,min}}, \quad
-Y_{\Gamma} = Y_{d,s} \left(Y_{\textrm{d,max}} - Y_{\textrm{d,min}}\right) + Y_{\textrm{d,min}}
+X_{\Gamma} = X_{\textrm{d,s}} \left(X_{\textrm{d,max}} - X_{\textrm{d,min}}\right) + X_{\textrm{d,min}}, \quad
+Y_{\Gamma} = Y_{\textrm{d,s}} \left(Y_{\textrm{d,max}} - Y_{\textrm{d,min}}\right) + Y_{\textrm{d,min}}
 ```
 Where the last entry is the exact quantile of interest 
 
@@ -133,7 +134,7 @@ X_{\alpha} = X_{\Gamma}[-1], \quad
 Y_{\alpha} = Y_{\Gamma}[-1]
 ```
 
-**Step 6** <br>
+**Step 7** <br>
 Compute the adjustment coefficient associated with the quantile of interest -
 ```math
 \textrm{if}\;\textrm{Y-dependent}\;\Rightarrow C_{\alpha} = \frac{Y_{\alpha}}{X_{\alpha}^m}