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Fix links linear_model
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‎docs/docs/notebooks/synthetic-examples/01_linear_model.md‎

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@@ -26,8 +26,6 @@ Hopefully, there is a quantity called **heterogeneity** that can be used to chec
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`Effector` provides five different feature effect methods, which are summarized in the table below. In all methods, setting `heterogeneity=True` the methods show the level of heterogeneity, along with the average effect.
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<center>
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| Method | Description | API in `Effector` | Paper |
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|-----------------------------------------|------------------------------------|------------------------------------------------------------------------------|-------------------------------------------------------------------|
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| [PDP](#partial-dependence-plot-pdp) | Partial Dependence Plot | [PDP]((./../../reference/#effector.pdp.DerivativePDP)) | [Friedman, 2001](https://statweb.stanford.edu/~jhf/ftp/trebst.pdf) | |
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| [SHAP](#shap-dependence-plot) | SHAP Dependence Plot | [SHAPDependence](./../../reference/#effector.shap_dependence.SHAPDependence) | [Lundberg et. al, 2017](https://arxiv.org/pdf/1705.07874.pdf) |
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</center>
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For the rest of the tutorial, we will use the following notation:
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| Symbol | Description |
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|------------------------------------------------------------|---------------------------------------------------------|
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| $f(\mathbf{x})$ | The black box model |
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| $\mathbf{x} = (x_s, x_c) = (x_1, x_2, ..., x_s, ..., x_D)$ | The input features |
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| $\mathbf{x}^{(i)} = (x_s^{(i)}, x_c^{(i)})$ | The $i$-th instance of the dataset |
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---
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## Dataset and Model
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In this example, we will use as black-box function, a simple linear model $y = 7x_1 - 3x_2 + 4x_3$. Since there are no interactions
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terms we expect **all** methods to provide the following feature effects and zero heterogeneity:
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| Feature | Feature Effect | Heterogeneity |
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| --- | --- | --- |
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| $x_1$ | $7x_1$ | 0 |
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| $x_2$ | $-3x_2$ | 0 |
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| $x_3$ | $4x_3$ | 0 |
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As dataset, we will generate $N=1000$ examples comming from the following distribution:
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For example, for $x_1$ the upper subplot shows a linear effect and the lower subplot confirms that the gradient is constantly $7$.
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`Effector` offers two alternatives for centering the ALE plot.
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| `centering` | Description | Formula |
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|---------------------------|----------------------------------------|-----------------------------------------------------------------------|
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| `False` or `zero_start` | Don't enforce any additional centering | c=0 |
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| `True` or `zero_integral` | Center around the $y$ axis | c=$\mathbb{E}_{x_s \sim \mathcal{U(x_{s,min},x_{s, max})}}[ALE(x_s)]$ |
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Let's see how centering works for $x_1$:
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As with the ALE, there are two alternatives for centering the ALE plot.
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| `centering` | Description | Formula |
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|---------------------------|----------------------------------------|-----------------------------------------------------------------------|
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| `False` or `zero_start` | Don't enforce any additional centering | c=0 |
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| `True` or `zero-integral` | Center around the $y$ axis | c=$\mathbb{E}_{x_s \sim \mathcal{U(x_{s,min},x_{s, max})}}[ALE(x_s)]$ |
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Let's see how this works for $x_1$:
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| `centering` | Description | Formula |
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|---------------------------|----------------------------------------|-----------------------------------------------------------------------|
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| `False` or `zero_start` | Don't enforce any additional centering | c=0 |
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| `True` or `zero-integral` | Center around the $y$ axis | c=$\mathbb{E}_{x_s \sim \mathcal{U(x_{s,min},x_{s, max})}}[ALE(x_s)]$ |
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Let's see how this works for $x_1$:
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the feature effect plot of the $s$-th feature `feature=s` is given with the table below.
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The argument `confidence_interval=True|False` indicates whether to plot the standard deviation of the instance-level effects as $\pm$ interval around the feature effect plot. Some methods also require the gradient of the model `model_jac: (N, D) -> (N, D)`.
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| Method | How to use |
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|---------------|----------------------------------------------------------------------------------------------------------------------------------------------|
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| PDP | [`effector.PDP(X, model).plot(feature, centering, confidence_interval)`]((./../../reference/#effector.pdp.PDP)) |
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| d-PDP | [`effector.DerivativePDP(X, model, model_jac).plot(feature, centering, confidence_interval)`](./../../reference/#effector.pdp.DerivativePDP) |
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| ALE | [`effector.ALE(X, model).plot(feature, centering, confidence_interval)`](./../../reference/#effector.ale.ALE) |
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| RHALE | [`effector.RHALE(X, model, model_jac).plot(feature, centering, confidence_interval)`](./../../reference/#effector.ale.RHALE) |
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</center>
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| PDP | [`effector.PDP(X, model).plot(feature, centering, confidence_interval)`](../../../api_docs/api_global/#effector.global_effect_pdp.PDP) |
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| d-PDP | [`effector.DerivativePDP(X, model, model_jac).plot(feature, centering, confidence_interval)`](../../../api_docs/api_global/#effector.global_effect_pdp.DerPDP) |
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| ALE | [`effector.ALE(X, model).plot(feature, centering, confidence_interval)`](../../../api_docs/api_global/#effector.global_effect_ale.ALE) |
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| RHALE | [`effector.RHALE(X, model, model_jac).plot(feature, centering, confidence_interval)`](../../../api_docs/api_global/#effector.global_effect_ale.RHALE) |
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```python

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