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Update regression approach powerlaw authored by Jamie Engelhardt Simon's avatar Jamie Engelhardt Simon
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......@@ -8,6 +8,7 @@ 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).
......@@ -88,7 +89,13 @@ Y_{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**
**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.
<br>
<div align="center">
![subset_error_band](uploads/b35fa634cc7dc99f39255bca21e6fa5c/subset_error_band.png){width=35%}&nbsp;&nbsp;
![quantile_exmaple_error](uploads/f6950315da8dfc9c3d3054c0dfd95cb5/quantile_exmaple_error.png){width=35%}
</div>
Compute the error/difference between the model and the data
......@@ -96,12 +103,6 @@ Compute the error/difference between the model and the data
E = Y_{\textrm{p}} - Y_{\textrm{d}}
```
<br>
<div align="center">
![subset_error_band](uploads/b35fa634cc7dc99f39255bca21e6fa5c/subset_error_band.png){width=35%}&nbsp;&nbsp;
![quantile_exmaple_error](uploads/f6950315da8dfc9c3d3054c0dfd95cb5/quantile_exmaple_error.png){width=35%}
</div>
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
......@@ -111,19 +112,25 @@ E = E\left[E > 0\right], \quad X_{s} = X_{s}[E>0], \quad Y_{s} = Y_{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}[\textrm{argsort}\left(E\right)\right], \quad Y_{s} = Y_{s}[\textrm{argsort}\left(E\right)\right]
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]
```
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]
```
The last entry in the scaled arrays are now the quantile of interest.
<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.
**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}}
```