From 35c248f522245af86505d211a1213a95726302e3 Mon Sep 17 00:00:00 2001 From: lucy liu Date: Fri, 16 Sep 2022 12:16:55 +1000 Subject: [PATCH] add blog --- .../2022-09-16-permutation-importance-shap.md | 404 ++++++++++++++++++ assets/images/posts_images/SHAP_overfit.png | Bin 0 -> 32802 bytes .../posts_images/permutation_importances.png | Bin 0 -> 11686 bytes .../permutation_importances_overfit.png | Bin 0 -> 17988 bytes assets/images/posts_images/shap_beeswarm.png | Bin 0 -> 35606 bytes assets/images/posts_images/shap_waterfall.png | Bin 0 -> 29855 bytes 6 files changed, 404 insertions(+) create mode 100644 _posts/2022-09-16-permutation-importance-shap.md create mode 100644 assets/images/posts_images/SHAP_overfit.png create mode 100644 assets/images/posts_images/permutation_importances.png create mode 100644 assets/images/posts_images/permutation_importances_overfit.png create mode 100644 assets/images/posts_images/shap_beeswarm.png create mode 100644 assets/images/posts_images/shap_waterfall.png diff --git a/_posts/2022-09-16-permutation-importance-shap.md b/_posts/2022-09-16-permutation-importance-shap.md new file mode 100644 index 0000000..684318b --- /dev/null +++ b/_posts/2022-09-16-permutation-importance-shap.md @@ -0,0 +1,404 @@ +--- +title: "Explaining machine learning models with permutation importance and SHAP" +date: September 16, 2022 + +categories: + - Technical +tags: + - Machine Learning + +featured-image: jml.png + +postauthors: + - name: Lucy Liu + website: https://twitter.com/lucyleeow + email: lliu@quansight.com + image: lucyliu.jpeg +--- + +
+ + {% include postauthor.html %} +
+ + +Understanding why a machine learning model produces an output is useful for a +variety of reasons; user trust, bias detection, debugging and opportunity +to improve the model. A number of methods have been developed to aid model +interpretation, which are particularly useful for complex models. In this post +we will compare two methods; permutation importance, implemented in +scikit-learn, and SHAP (**SH**apley **A**dditive ex**P**lanations). + +## Permutation importance + +Permutation importance uses the decrease in model score, after shuffling +the values of a feature, to measure how important each feature is. It can +be calculated using the scikit-learn function +[`permutation_importance`](https://scikit-learn.org/stable/modules/generated/sklearn.inspection.permutation_importance.html#sklearn.inspection.permutation_importance): + +```python +import matplotlib.pyplot as plt +import pandas as pd + +from sklearn.datasets import fetch_california_housing +from sklearn.ensemble import RandomForestRegressor +from sklearn.inspection import permutation_importance +from sklearn.model_selection import train_test_split + +# Only use a subset of the california housing dataset to speed up +# computation +cal_housing = fetch_california_housing() +y = cal_housing.target[::10] +X = pd.DataFrame(data=cal_housing.data[::10, :], columns=cal_housing.feature_names) +X.head() + +# Split data into training and testing subsets +X_train, X_test, y_train, y_test = train_test_split(X, y, random_state=7) + +# Fit the model with our training subset +reg = RandomForestRegressor(max_depth=10, n_jobs=2, random_state=7) +reg.fit(X_train, y_train) + +# Calculate permutation importances on the test subset using the default score +# metric of `RandomForestRegressor`: R² +perm_import = permutation_importance( + reg, X_test, y_test, n_repeats=10, random_state=42, n_jobs=2 +) +sorted_idx = perm_import.importances_mean.argsort() + +# Plot decrease in R² value for each permutation +fig, ax = plt.subplots() +ax.boxplot( + perm_import.importances[sorted_idx].T, + vert=False, + labels=X.columns[sorted_idx], +) +ax.set_title("Permutation Importances") +fig.tight_layout() +plt.show() +``` + +![Boxplot of permutation importances](/assets/images/posts_images/permutation_importances.png) + +The above plot shows that `MedInc` (median income) causes by far the biggest +drop in R² score whereas `AveBedrms` and `Population` seem to have +almost no effect. + +**Considerations** + +* Permutation importance is linked to the score metric used, thus selecting + an appropriate metric for your needs is essential. +* Permutation importance can be calculated with either the training or + testing subset and each provides different information about the model. + Using the training subset shows what the model has learned to use for + making predictions. Using the testing subset shows what is actually useful + when making predictions on novel data. Features that the model is able to + use to overfit on the training data will appear more important when using + the training data than the testing data. This can be seen in + [this scikit-learn example](https://scikit-learn.org/stable/auto_examples/inspection/plot_permutation_importance.html) + where random features are added into the dataset. +* Permutation importance depends on the true `y` to quantify the ability + of features to contribute to correct predictions, contrary to SHAP which + quantifies the ability of features to contribute to changes to the + decision function, irrespective of whether this improves the accuracy of + the model. + +**Advantages** + +* Permutation importances are easy to interpret and provide information on + the global importance of each feature. +* Permutations can be performed on original feature values or transformed + feature values. Consequently importances would be either interpreted + in the original feature space or the transformed feature space. + In scikit-learn this is achieved by either using an estimator or a + [`pipeline`](https://scikit-learn.org/stable/modules/generated/sklearn.pipeline.Pipeline.html) + as input to the `permutation_importance` function. + Only estimators are accepted as arguments in the SHAP package, meaning + importance values would need to be interpreted in the transformed + feature space. +* Permutation importances are error ratios, where 1 means no + difference between error with and without permutation. This means + these values are comparable between different models and problems. + +**Disadvantages** + +* Feature permutation destroys all dependency between features. This means + the permutation importance value represents the combination of + the main effect and any interaction effects with other features. Take + for example two features that interact with each other. The interaction + effects would be 'accounted for' twice, once in the importance value of + each feature. This means that the decrease in score from permuting each + feature would not add up to equal the overall decrease in score from + permuting all features, unless there were no interaction effects between + features. +* Permutation importance assumes independence between features. The effect + is that the 'importance' is split between correlated features. See + [this scikit-learn example](https://scikit-learn.org/stable/auto_examples/inspection/plot_permutation_importance_multicollinear.html) + for a demonstration of this. +* Permutation can result in unrealistic data points. +* Permutation importances can only be used to compute a global "explanation", + a decomposition of the impact of different features for a full dataset. + Unlike SHAP, it cannot provide local "explanations" for individual samples. + +## SHAP + +Shapley values are an idea from coalition game theory to fairly attribute +value to players of a cooperative game. Applying this method to model +explanations, the features become the players and shapley values attribute the +prediction output of the model amongst the features. It is calculated as the +average marginal contribution of each feature, across all possible feature +combinations. These values satisfy a number of good properties and are thus +deemed a fair way to 'split' the prediction output between the features. + +SHAP are shapley values of a conditional expectation function of the original +model, attributing to each feature the change in model prediction when +conditioning on that feature. They can be interpreted as how much a +feature contributes to the prediction, for a specific sample, compared to the +average prediction for the dataset. SHAP values are represented as a linear model +(additive feature attribution method) of feature subsets. Practically, the +predicted outcome is evaluated for all possible subsets of features, with and +without the feature in question. This allows SHAP to account for interaction +effects between features but makes computation very expensive as the number of +possible feature subsets increases exponentially. A number of +approximation methods have been proposed, including the model-agnostic +KernelSHAP and model-specific TreeSHAP. KernelSHAP combines the concepts of +LIME (local surrogate model of feature importance) and shapley values. +It calculates the predictions for a sample when different subsets of +the features are 'missing', simulating missingness via a background +value for that feature. These predictions are then used to +fit a linear model whose predictions match that of the original model as +closely as possible. The coefficients of the linear 'explanation' model are +the SHAP values. TreeSHAP is designed for tree-based models and uses +the difference between the conditional expectation of feature subsets with and +without the feature of interest to estimate SHAP values. + +Several SHAP approximation methods are implemented in the Python library +`SHAP `_. SHAP methods implementated here +support models from various packages, including scikit-learn, XGBoost and Keras. +Below is an example using the SHAP package and a scikit-learn estimator. + +```python +import shap + +# By default `Explainer` will select the best algorithm to estimate SHAP, +# given the model passed. Below we are passing a tree based model so the +# algorithm used will be TreeSHAP +explainer = shap.Explainer(reg) +# Calculate SHAP values for test data set +shap_values = explainer(X_test) +``` + +`shap_values` is an 'explanation object'. Some useful attributes include: + +* `values`: the SHAP values +* `base_values`: the global mean prediction +* `data`: the data the SHAP values have been calculated for + +SHAP values for each feature and each data point of `X_test` are calculated. +We can see that `values` is the same shape as `X_test`: + +```python +print(f"SHAP values shape: {shap_values.values.shape}") +print(f"X_test shape: {X_test.shape}") +``` + +gives: + +``` +SHAP values shape: (516, 8) +X_test shape: (516, 8) +``` + +Exploring the SHAP values of the first data point using `shap.plots.waterfall`: + +![Waterfall plot of SHAP values](/assets/images/posts_images/shap_waterfall.png) + +The SHAP values for each feature 'pushes' the mean prediction (the expected +value) higher (positive SHAP value) or lower (negative SHAP value). +The SHAP values for one data point should sum to the difference between the +prediction output by our model `reg` and the `base_value`: + +```python +prediction = reg.predict(X_test.iloc[[0]]) +print( + "Difference between model prediction and base value: " + f"{(prediction - shap_values.base_values[0])[0]:.2f}" +) +print(f"Sum of SHAP values: {shap_values[0].values.sum():.2f}") +``` + +``` +Difference between model prediction and base value: -1.13 +Sum of SHAP values: -1.13 +``` + +We can also plot a summary of all the SHAP values: + +![Waterfall plot of SHAP values](/assets/images/posts_images/shap_beeswarm.png) + +In the plot above, each dot represents the SHAP value of one feature and one +sample. The features are ordered from most important (highest mean absolute +SHAP value) at the top to least important at the bottom. Note that the dots +cluster around 0 (no contribution) more and more as you go down. The dots are +also colored by the value of the feature. Looking at the `MedInc` feature, +you can clearly see that as feature value increases, SHAP value generally +increases as well. + +**Advantages** + +* This method allows you to compute feature importances for individual + samples (local explanations). This can then be extended to global + explanations by statistically summarizing local explanations in some + way (e.g., taking average) +* Interaction effects are accounted for, unlike in permutation importances. + For each sample, the contribution of each feature add up to the overall + prediction, as shown above. + +**Disadvantages** + +* SHAP are very computationally expensive, especially when there + are a large number of features (as possible combinations of + features exponentially increases). SHAP methods do differ in their computation + times though, for example TreeSHAP is able to compute SHAP values in + polynomial time while KernelSHAP does it in exponential time. +* SHAP also assumes independence between features. Similar to + permutation importance, the result is that 'contributions' will be split + between correlated features. +* Certain SHAP methods (e.g., KernalSHAP) rely on evaluating the model on + data points outside of the actual data distributon, which can be unrealistic. +* For certain feature distributions, the SHAP values of non-additive models may + uninformative. +* SHAP does not take into account model performance. It would thus + be valuable to first assess model performance before investigating + feature importances using this method. + +## Comparison + +In practice, both methods will generally order features similarly in terms +of their importance in a model. If you compare with the permutation importance +plot with the beeswarm SHAP plot, you will notice that the order of features is +roughly the same. However, there are some important differences and practical +considerations between the methods summarized below: + +| SHAP | Feature importance | +|----------------------------------------------|------------------------------------------------------| +| Model-agnostic | Model-agnostic | +| Provides local and global importances | Provides global importance only | +| Assumes independence between features | Assumes independence between features | +| Accounts for interaction between features | Does not account for interaction
between features | +| Computationally expensive | Less computationally expensive | +| Does not take model performance into account | Takes model performance into account | + +The last point is of particular interest. SHAP values quantify the contribution +of each feature to the prediction output, irrespective of whether this improves +the model accuracy, unlike permutation importances. This can be seen in the +example below where we add some noise to our target (`y`) and allow our model +to overfit. For permutation importances, there is a clear difference between +the importance (reduction in R²) between the test set and the train set. This +is because the model was allowed to overfit, resulting in better performance +on the train set than the test set, leading to higher importance values. + +```python +# Add noise +y_noise = y + np.random.normal(0, y.std(), y.shape) +X_train, X_test, y_train, y_test = train_test_split(X, y_noise, random_state=7) + +# fit the model, increasing `max_depth` to overfit the model +reg = RandomForestRegressor(max_depth=100, n_jobs=2, random_state=7) +reg.fit(X_train, y_train) + +# calculate permutation importances on the test subset +perm_import_test = permutation_importance( + reg, X_test, y_test, n_repeats=10, random_state=42, n_jobs=2 +) +sorted_idx_test = perm_import_test.importances_mean.argsort() +# calculate permutation importances on the train subset +perm_import_train = permutation_importance( + reg, X_train, y_train, n_repeats=10, random_state=42, n_jobs=2 +) + +# Function for creating side by side boxplots +def plot_boxplots(test_data, train_data, method, legend_loc): + fig, ax = plt.subplots(figsize=(10, 7)) + # Plot test importances + box_test = ax.boxplot( + test_data, + positions=1 + np.arange(test_data.shape[1]) * 2.2, + vert=False, + patch_artist=True, + manage_ticks=False, + medianprops={"color": "white", "linewidth": 0.5}, + boxprops={"facecolor": "blue"}, + ) + # Plot train importances + data = shap_values_train.values + box_train = ax.boxplot( + train_data, + positions=1.6 + np.arange(train_data.shape[1]) * 2.2, + vert=False, + manage_ticks=False, + patch_artist=True, + medianprops={"color": "white", "linewidth": 1}, + boxprops={"facecolor": "darkgoldenrod"}, + ) + ax.set_title(f"{method} (noisy target & overfit model)", fontsize=15) + plt.yticks( + 1.4 + np.arange(data.shape[1]) * 2.2, X.columns[sorted_idx_test], fontsize=12 + ) + ax.legend( + [box_train["boxes"][0], box_test["boxes"][0]], + ["Test", "Train"], + loc=legend_loc, + fontsize=15, + ) + fig.tight_layout() + plt.show() + +# Plot boxplot +plot_boxplots( + perm_import_test.importances[sorted_idx_test].T, + perm_import_train.importances[sorted_idx_test].T, + "Permutation Importances", + "lower right", +) +``` + +![Side-by-side boxplot of permutation importances in overfit model](/assets/images/posts_images/permutation_importances_overfit.png) + +If we use the overfit model and calculate SHAP values for the test and train +set, we can see that SHAP values are very similar between the test and train +sets. It is advisable to evaluate the performance of your model before +calculating SHAP values. + +```python +# Calculate SHAP value for train and test sets +explainer = shap.Explainer(reg) +shap_values_test = explainer(X_test) +shap_values_train = explainer(X_train) +# Plot boxplots +plot_boxplots( + shap_values_test.values, shap_values_train.values, "SHAP", "upper right" +) +``` + +![Side-by-side boxplot of SHAP values in overfit model](/assets/images/posts_images/SHAP_overfit.png) + +## References + +* Lundberg, S. M., & Lee, S. I. (2017). A unified approach to interpreting model + predictions. Advances in neural information processing systems, 30. +* Lundberg, S. M., Erion, G. G., & Lee, S. I. (2018). Consistent individualized + feature attribution for tree ensembles. arXiv preprint arXiv:1802.03888. +* Sundararajan, M., & Najmi, A. (2020, November). The many Shapley values for + model explanation. In International conference on machine learning + (pp. 9269-9278). PMLR. +* Aas, K., Jullum, M., & Løland, A. (2021). Explaining individual predictions + when features are dependent: More accurate approximations to Shapley values. + Artificial Intelligence, 298, 103502. +* Kumar, I. E., Venkatasubramanian, S., Scheidegger, C., & Friedler, S. + (2020, November). Problems with Shapley-value-based explanations as feature + importance measures. In International Conference on Machine Learning + (pp. 5491-5500). PMLR. +* Molnar, Christoph. "Interpretable machine learning. 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