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).
@@ -57,7 +47,7 @@ There exist a multitude of methods as to identify these limits such as assuming
 <br>
 Set quantile of interest 
 ```math
-\alpha_{0.75} = 0.75, \quad \alpha_{0.95} = 0.95
+\alpha = \alpha
 ```
 
 **Step 2**
@@ -65,7 +55,7 @@ Set quantile of interest
 Compute error limit
 
 ```math
-\Gamma_{0.75} = 1 - \alpha_{0.75}, \quad \Gamma_{0.95} = 1 - \alpha_{0.95}
+\Gamma = 1 - \alpha
 ``` 
 
 **Step 3**
@@ -73,8 +63,7 @@ Compute error limit
 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)
+I_{\alpha} = \textrm{int}\left(\textrm{round}\left(k\cdot\Gamma\right)\right), \quad 
 ```
 
 where $k$ is the number of points in the data-series.
@@ -85,8 +74,13 @@ 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 and aid in faster convergence and accuracy.
 
 ```math
-X_{s} = \frac{X_i - X_{\textrm{min}}}{X_{\textrm{max}} - X_{\textrm{min}}}, \quad
-Y_{s} = \frac{Y_i - Y_{\textrm{min}}}{Y_{\textrm{max}} - Y_{\textrm{min}}}
+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}}}
+```
+
+```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}}}
 ```
 
 where the lower index s is an abbreviation for scaled.
@@ -102,34 +96,34 @@ The purpose is to sort the difference between the prediction and the data and de
 Compute the error/difference between the model and the data 
 
 ```math
-E = Y_{\textrm{p}} - Y_{\textrm{d}}
+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.
 
 ```math
-E = E\left[E > 0\right], \quad X_{s} = X_{s}[E>0], \quad Y_{s} = Y_{s}[E>0]
+E = E\left[E > 0\right], \quad X_{d,s} = X_{d,s}[E>0], \quad Y_{d,s} = Y_{d,s}[E>0]
 ```
 
 Sort the data based on the error
  
 ```math
 E = E\left[\textrm{argsort}\left(E\right)\right],
-\quad X_{s} = X_{s}\left[\textrm{argsort}\left(E\right)\right],
-\quad Y_{s} = Y_{s}\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]
 ```
 Extract the data within the desired quantile range
 ```math
 E = E\left[:-I_{\alpha}\right],
-\quad X_{s} = X_{s}\left[:-I_{\alpha}\right],
-\quad Y_{s} = Y_{s}\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]
 ```
 **Step 6** <br>
 Renormalize the data.
 
 ```math
-X_{\Gamma} = X_s \left(X_{\textrm{d,max}} - X_{\textrm{d,min}}\right) + X_{\textrm{d,min}}, \quad
-Y_{\Gamma} = Y_s \left(Y_{\textrm{d,max}} - Y_{\textrm{d,min}}\right) + Y_{\textrm{d,min}}
+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}}
 ```
 Where the last entry is the exact quantile of interest 
 
@@ -140,16 +134,14 @@ Y_{\alpha} = Y_{\Gamma}[-1]
 
 **Step 6** <br>
 Compute the adjustment coefficient associated with the quantile of interest -
-
-if Y dependent
 ```math
-C_{\alpha} = \frac{Y_{\alpha}}{X_{\alpha}^m}
+\textrm{if}\;\textrm{Y-dependent}\;\Rightarrow C_{\alpha} = \frac{Y_{\alpha}}{X_{\alpha}^m}
 ```
-if X dependent
 ```math
-C_{\alpha} = \frac{X_{\alpha}}{Y_{\alpha}^m}
+\textrm{if}\;\textrm{X-dependent}\;\Rightarrow C_{\alpha} = \frac{X_{\alpha}}{Y_{\alpha}^m}
 ```
 
+The end results for an X- or Y dependent fit using $\alpha=0.75$ and $\alpha=0.95$ are illustrated below