home/regression-approach.md

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 title: Regression approach and Tolerance bands - power law
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 ## Dependency
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-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).
+It is assumed that the data can be described by a power law, expressed as Y or X-dependent (referring to the second and first axes on a fatigue curve).
 
 ```math
 Y = CX^m \leftrightarrow X = \left(\frac{Y}{C}\right)^{\frac{1}{m}}  
@@ -23,8 +14,7 @@ X = CY^m \leftrightarrow Y = \left(\frac{X}{C}\right)^{\frac{1}{m}}
 Where $m$ and $C$ are the power slope- and adjustment coefficient, respectively.
 
 ## Fitting procedure
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-In using the power-law as a prediction function, the model can be fitted in logspace as follows.
+Using the power-law as a prediction function, the model can be fitted in log10-space as follows.
 
 ```math
 X_{\textrm{log}} = \textrm{log10}\left(X\right), \quad Y_{\textrm{log}} = \textrm{log10}\left(Y\right) 
@@ -42,16 +32,17 @@ C = 10^{\left(Y_\text{mean} - m X_\text{mean}\right)}
 ```
  
 ## Tolerance bands
-A tolerance band states that a certain percentage of the data lies within or above the prediction statement. It is typically of interest to identify the lower-limits in which 75 and 95 percent of the data are above the prediction statement to estimate a conservative measure of the blade lifetime. Below image illustrates a set of data points $\left(X_\textrm{d},Y_\textrm{d}\right)$ with a mean power curve $\left(X_\textrm{p},Y_\textrm{p}\right)$.
+A tolerance band states that a certain percentage of the data lies within or above the prediction statement. It is typically interesting to identify the lower limits in which 75- and 95 % of the data are above the prediction statement, to estimate a conservative measure of the blade lifetime.
+The image below illustrates data points $\left(X_\textrm{d},Y_\textrm{d}\right)$ with a mean power curve $\left(X_\textrm{p},Y_\textrm{p}\right)$.
 
 <br>
 <div align="center">
 ![power_curve_example](uploads/77a961d851186f96dbe42b8e792e2d86/power_curve_example.png){width=45%}
 </div>
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-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
+Several methods exist to identify these limits, such as assuming the data follows a normal- or Weibull distribution. However, in this scenario, the distribution's upper- and lower quantiles are used to identify said limits. The following steps explain the algorithm step by step.
 <br>
 <br>
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 **Step 1**
 <br>
 Set quantile of interest 
@@ -72,7 +63,7 @@ Compute error limit
 Employ the interval on the active data-range
 
 ```math
-I_{\alpha} = \textrm{int}\left(\textrm{round}\left(k\cdot\Gamma\right)\right), \quad 
+I_{\alpha} = \textrm{int}\left(\textrm{round}\left(k\cdot\Gamma\right)\right)
 ```
 
 where $k$ is the number of points in the data-series.
@@ -80,7 +71,7 @@ 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, it also aids in faster convergence and accuracy.
+Perform _min-max_ normalization. This ensures that the features used by the model have similar scales and aids in faster convergence and accuracy.
 
 ```math
 X_{\textrm{d,s}} = \frac{X_\textrm{d} - X_{\textrm{d,min}}}{X_{\textrm{d,max}} - X_{\textrm{d,min}}}, \quad
@@ -95,7 +86,7 @@ Y_{\textrm{p,s}} = \frac{Y_\textrm{p} - Y_{\textrm{d,min}}}{Y_{\textrm{d,max}} -
 where the lower index s is an abbreviation for scaled.
 
 **Step 5** <br>
-The purpose is to sort the difference between the prediction and the data and determine the error's associated with the quantile of interest. The procedure is illustrated below and the underlining steps are listed.
+The purpose is to sort the difference between the prediction and the data and determine the errors associated with the quantile of interest. The procedure is illustrated below, and the underlying steps are listed.
 <br>
 <div align="center">
 ![subset_error_band](uploads/b35fa634cc7dc99f39255bca21e6fa5c/subset_error_band.png){width=35%}&nbsp;&nbsp;