home/regression-approach.md

@@ -4,6 +4,7 @@ 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).
@@ -24,6 +25,18 @@ In using the power-law as a prediction function, the model can be fitted in logs
 ```math
 X_{\textrm{log}} = \textrm{log10}\left(X\right), \quad Y_{\textrm{log}} = \textrm{log10}\left(Y\right) 
 ```
+The power exponent or slope can be described as
+
+```math 
+m = \frac{\sum_{\text{i} = 0}^{\text{n}} \left(X_\text{i}-X_\text{mean}\right)\left(Y_\text{i}-Y_\text{mean}\right)}{\sum_{\text{i} = 0}^{\text{n}} \left(X_\text{i}-X_\text{mean}\right)^2}
+```
+
+and the adjustment coefficient is
+
+```math
+C = 10^{\left(Y_\text{mean} - m X_\text{mean}\right)}
+```
+